If I graft a scion from one apple tree onto another mature tree, how long will it take for the graft to begin bearing fruit?
Two years, it needs a growing season to establish, will set leaf buds the first winter and will start setting flower buds the second winter. Sometimes you'll get an early blossom, but you should cut it off early so it doesn't stress (physically or nutritionally) the graft.
If you splice graft, whip and tongue graft, or bud (or any of the many similar grafts) onto stems that are under 1/2" in diameter, it should produce budwood in 2-3 years.
Generally with a diameter larger than that, at least with apple, you'll be rind grafting or cleft grafting. These will put out lots of growth in the next few years, to rebuild the removed mass, and often don't fruit for 5-7 years. Dwarf trees are always faster.